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I am self-studying for an actuarial exam, and I encountered the following in my text:

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The author states that if $PV_{t, T}\text{(Divs)} < K(1 - e^{-r(T - t)})$, early exercise is not rational.

That made me wonder if the converse is true: If $PV_{t, T}\text{(Divs)} \geq K(1 - e^{-r(T - t)})$, then early exercise may be rational.

But in an exercise, the author seems to suggest that there is more to it than that:

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The author seems to suggest that we have to also consider the value of the put. I don't understand why we have to consider the value of the put in this example, yet we did not consider the value of the put in the statement underlined in red.

I'm looking for clarification on this.

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If $PV_{t, T}(\text{Divs}) \ge K\big(1-e^{-r(T-t)}\big)$, since $P_{Eur}(S_t, K, T-t) >0$, the identity \begin{align*} C_{Eur}(S_t, K, T-t) = P_{Eur}(S_t, K, T-t) + (S_t-K) -PV_{t, T}(\text{Divs}) +K\big(1-e^{-r(T-t)}\big), \end{align*} implies that \begin{align*} C_{Eur}(S_t, K, T-t) > (S_t-K). \end{align*} That is, it is not rationale to exercise the option at time $t$ in this case. Note that the conclusion also depends on the statement that "The put must be worth at least zero", which you should also highlight.

$$ $$ Note that, it may be rationale to exercise the option at $t$, instead of the maturity $T$, only if \begin{align*} C_{Eur}(S_t, K, T-t) \le (S_t-K). \end{align*} From the above identity, this is equivalent to \begin{align*} P_{Eur}(S_t, K, T-t) -PV_{t, T}(\text{Divs}) +K\big(1-e^{-r(T-t)}\big) \le 0. \end{align*} That is, the put option price is indeed taking into consideration. For the Quiz you provided, since \begin{align*} P_{Eur}(S_t, K, T-t) -PV_{t, T}(\text{Divs}) +K\big(1-e^{-r(T-t)}\big) &= 0.82 -2.9851 +1.4049 \\ &< 0, \end{align*} it may be rationale to exercise this option early.

$$$$ Here, we say that it may be rationale to exercise early. However, this does not mean it is optimal to exercise at time $t$, as we only considered two possible exercise times $t$ and $T$. In general, we need to take all possible stopping times, $\tau$, ranged from $t$ to $T$ into consideration. That is, we need to solve an optimal stopping problem.

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